对称系统的耗散性分析与控制
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摘要
对称系统是一类具有特殊结构的系统,是有着广泛应用背景的动力系统。如电路系统、电子网络系统、电力网系统、大型的空间结构系统、弹性材料系统和化学反应系统等。耗散性理论自20世纪70年代提出以来,在系统稳定性研究过程中起到重要的作用。耗散性理论从能量的角度提出了一种控制系统设计与分析的思想,不仅在控制理论方面有着重要的作用,而且在许多实际系统,如机器人系统、电力系统、化工过程的控制研究方面也是一种有效方法。由于它们在动力学系统、控制理论和控制工程实践中的广泛应用,近年来,对称系统和耗散性理论的研究引起了国内外众多学者的关注,并且己经取得了长足的进展。
本论文针对当前对称系统、耗散性理论的研究现状,重点研究了线性对称系统、线性时滞对称系统、线性广义对称系统的耗散分析与控制以及鲁棒耗散分析与控制问题。具体的研究内容如下:
(1)研究了状态空间对称系统的耗散性分析与控制问题。首先,对于状态空间对称系统,应用矩阵代数知识和线性矩阵不等式方法,给出了系统耗散的充要条件及耗散指标的显式表达式;其次,我们研究了线性状态空间对称系统的镇定问题,基于静态输出反馈控制器,给出系统镇定的充要条件,并给出了保证系统镇定的控制器的设计范围;最后研究了线性状态空间对称系统的耗散性控制问题。在分析结论的基础之上,给出了闭环系统的耗散指标的显式表达式,同时也给出了控制器增益的参数化显式表示。
(2)研究了状态空间对称时滞系统的耗散性分析与控制问题。首先对连续时间状态空间对称时滞系统,研究了系统的耗散性分析、镇定与控制问题。运用矩阵代数工具,给出了系统耗散指标的显式表达式、系统镇定的控制器的参数化设计范围、闭环系统耗散指标的参数化显示表达式以及静态输出反馈控制器增益的参数化设计方法;其次,对离散时间状态空间对称时滞系统,研究了耗散性分析,镇定与耗散性控制问题,得到了相应的结论。
(3)研究了广义对称系统的耗散性分析问题。分别针对连续时间和离散时间状态空间对称广义系统,我们给出了系统H∞范数计算的显式表达式。所得表达式仅仅与系统的参数矩阵显式相关,在计算上具有很大的优势。
(4)研究了线性系统基于PI/PID控制器的耗散性分析与控制问题。本文分别设计了类似于静态状态反馈的PI控制器,静态输出反馈的PI控制器以及类似于动态输出反馈的PID控制,对系统的耗散性控制问题做了研究。在所得结果基础之上,对系统参数带有不确定性的线性系统,得到了鲁棒耗散控制器的参数化显式表示。
本文以线性对称系统,线性时滞对称系统,广义对称系统和一般线性系统为研究对象进行研究的,所采用的方法和所给的结论均具有一般性。
Symmetric system is a system with a special structure and widely used inthe dynamical system. Examples of such systems include circuit systems, elec-trical and power networks, structural systems, viscoelastic materials, chemicalreactions and so on. Dissipative theory plays an important role in the stabilityresearch of control systems since it was brought forward in 1970s. From theviewpoint of energy, a kind of analyze and design idea is put forward in thedissipative theory, which not only playing an important role in the controltheory aspect, but also in many real systems, such as robot systems, powersystems, chemical systems and so forth. In view of the extensive application indynamic systems, control theory and control engineering practice in the pastyears, symmetric system and dissipative theory have received considerable at-tention and made great progress.
Based on the current research situation of symmetry system and dissi-pative theory, this thesis focuses on the problem of dissipative control androbust dissipative for linear symmetric systems, linear delayed symmetric sys-tems, linear descriptor symmetric systems. The main contents of the thesisare organized as follows:
(1) The dissipative analysis and control problems for state-space symmet-ric system are discussed. Firstly, for state-space symmetric system, based onmatrix algebra tools and linear matrix inequalities, we give the necessary andsu?cient conditions for dissipative system and the explicit expression for dis-sipative index. Then, we consider the stabilization problems for state-spacesymmetric system and design a static output feedback controller such that theclosed-loop system be stable. The necessary and su?cient conditions for sta-bilization system and the range of controller are given. Last we consider thedissipative control problems for linear state-space symmetric system. Fromthe analysis results, the explicit expressions of dissipative index and gain of static output feedback controllers are given.
(2) The dissipative analysis and control problems for state-space sym-metric time-delay system are investigated. At first, for state-space symmetricdelayed system, based on matrix algebra tools and linear matrix inequalities,we consider the dissipative analysis problems, stabilization problems and dis-sipative control problems. With regard to these problems, we give the explicitexpression of dissipative index and the parametric representation of staticoutput feedback controller gain. Then to the discrete time delay systems withstate-space symmetry, we obtain the similar results.
(3) The dissipative analysis problems for state-space symmetric descriptorsystem are studied. we give an explicit expression of H∞norm computationfor continuous-time symmetric descriptor system and discrete-time symmetricdescriptor system respectively. The expressions only relate to the parametermatrix of descriptor systems and have advantages in computation.
(4) The dissipative analysis and control problems for linear systems viaPI/PID controllers are considered. For linear time invariant systems, we designa PI controller which is similar to the static state feedback controller and aPI controller which is similar to the static output feedback controller anda PID controller which is similar to the dynamic output feedback controllerto investigate the dissipative control problems. On the basis of dissipativeanalysis and control results, we also discuss robust dissipative control problemsfor the linear time invariant systems with uncertainty, and give the parametricrepresentation for the PI/PID controller gains.
In the thesis, the research objects consist of linear symmetric systems,linear time-delay symmetric systems, descriptor symmetric system and generallinear system. The methods and the conclusions are given more general.
本论文针对当前对称系统、耗散性理论的研究现状,重点研究了线性对称系统、线性时滞对称系统、线性广义对称系统的耗散分析与控制以及鲁棒耗散分析与控制问题。具体的研究内容如下:
(1)研究了状态空间对称系统的耗散性分析与控制问题。首先,对于状态空间对称系统,应用矩阵代数知识和线性矩阵不等式方法,给出了系统耗散的充要条件及耗散指标的显式表达式;其次,我们研究了线性状态空间对称系统的镇定问题,基于静态输出反馈控制器,给出系统镇定的充要条件,并给出了保证系统镇定的控制器的设计范围;最后研究了线性状态空间对称系统的耗散性控制问题。在分析结论的基础之上,给出了闭环系统的耗散指标的显式表达式,同时也给出了控制器增益的参数化显式表示。
(2)研究了状态空间对称时滞系统的耗散性分析与控制问题。首先对连续时间状态空间对称时滞系统,研究了系统的耗散性分析、镇定与控制问题。运用矩阵代数工具,给出了系统耗散指标的显式表达式、系统镇定的控制器的参数化设计范围、闭环系统耗散指标的参数化显示表达式以及静态输出反馈控制器增益的参数化设计方法;其次,对离散时间状态空间对称时滞系统,研究了耗散性分析,镇定与耗散性控制问题,得到了相应的结论。
(3)研究了广义对称系统的耗散性分析问题。分别针对连续时间和离散时间状态空间对称广义系统,我们给出了系统H∞范数计算的显式表达式。所得表达式仅仅与系统的参数矩阵显式相关,在计算上具有很大的优势。
(4)研究了线性系统基于PI/PID控制器的耗散性分析与控制问题。本文分别设计了类似于静态状态反馈的PI控制器,静态输出反馈的PI控制器以及类似于动态输出反馈的PID控制,对系统的耗散性控制问题做了研究。在所得结果基础之上,对系统参数带有不确定性的线性系统,得到了鲁棒耗散控制器的参数化显式表示。
本文以线性对称系统,线性时滞对称系统,广义对称系统和一般线性系统为研究对象进行研究的,所采用的方法和所给的结论均具有一般性。
Symmetric system is a system with a special structure and widely used inthe dynamical system. Examples of such systems include circuit systems, elec-trical and power networks, structural systems, viscoelastic materials, chemicalreactions and so on. Dissipative theory plays an important role in the stabilityresearch of control systems since it was brought forward in 1970s. From theviewpoint of energy, a kind of analyze and design idea is put forward in thedissipative theory, which not only playing an important role in the controltheory aspect, but also in many real systems, such as robot systems, powersystems, chemical systems and so forth. In view of the extensive application indynamic systems, control theory and control engineering practice in the pastyears, symmetric system and dissipative theory have received considerable at-tention and made great progress.
Based on the current research situation of symmetry system and dissi-pative theory, this thesis focuses on the problem of dissipative control androbust dissipative for linear symmetric systems, linear delayed symmetric sys-tems, linear descriptor symmetric systems. The main contents of the thesisare organized as follows:
(1) The dissipative analysis and control problems for state-space symmet-ric system are discussed. Firstly, for state-space symmetric system, based onmatrix algebra tools and linear matrix inequalities, we give the necessary andsu?cient conditions for dissipative system and the explicit expression for dis-sipative index. Then, we consider the stabilization problems for state-spacesymmetric system and design a static output feedback controller such that theclosed-loop system be stable. The necessary and su?cient conditions for sta-bilization system and the range of controller are given. Last we consider thedissipative control problems for linear state-space symmetric system. Fromthe analysis results, the explicit expressions of dissipative index and gain of static output feedback controllers are given.
(2) The dissipative analysis and control problems for state-space sym-metric time-delay system are investigated. At first, for state-space symmetricdelayed system, based on matrix algebra tools and linear matrix inequalities,we consider the dissipative analysis problems, stabilization problems and dis-sipative control problems. With regard to these problems, we give the explicitexpression of dissipative index and the parametric representation of staticoutput feedback controller gain. Then to the discrete time delay systems withstate-space symmetry, we obtain the similar results.
(3) The dissipative analysis problems for state-space symmetric descriptorsystem are studied. we give an explicit expression of H∞norm computationfor continuous-time symmetric descriptor system and discrete-time symmetricdescriptor system respectively. The expressions only relate to the parametermatrix of descriptor systems and have advantages in computation.
(4) The dissipative analysis and control problems for linear systems viaPI/PID controllers are considered. For linear time invariant systems, we designa PI controller which is similar to the static state feedback controller and aPI controller which is similar to the static output feedback controller anda PID controller which is similar to the dynamic output feedback controllerto investigate the dissipative control problems. On the basis of dissipativeanalysis and control results, we also discuss robust dissipative control problemsfor the linear time invariant systems with uncertainty, and give the parametricrepresentation for the PI/PID controller gains.
In the thesis, the research objects consist of linear symmetric systems,linear time-delay symmetric systems, descriptor symmetric system and generallinear system. The methods and the conclusions are given more general.
引文
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